AdjacencyGraph

A Graph built from an n x n 0/1 adjacency matrix.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageAdjacencyGraph✓↗

AdjacencyGraph(matrix)vertices 1..n; a SYMMETRIC matrix gives an undirected graph (one edge per i < j with a nonzero entry), anything else gives a directed graph (one edge per nonzero (i, j), including self-loops).

AdjacencyGraph(vertices, matrix)the same, with vertices labelled by the given list instead of 1..n.

Typed signatures

Type	Package	Applies
(any, any?) -> value	combinatorics	always

Domain: Collections

Examples

Basic #

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In$\mathrm{EdgeList}(\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3)\bigr\rbrack$

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In$\mathrm{EdgeList}(\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\mathrm{DirectedEdge}(1, 2)\bigr\rbrack$

A symmetric 0/1 matrix with zero diagonal gives a triangle

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(2, 3)\bigr\rbrack)$

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 0\bigr\rbrack, \bigl\lbrack0, 0, 1\bigr\rbrack, \bigl\lbrack1, 0, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{DirectedEdge}(1, 2), \mathrm{DirectedEdge}(2, 3), \mathrm{DirectedEdge}(3, 1)\bigr\rbrack)$

Scope #

Vertices default to $1,\dots,n$

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In$\mathrm{VertexList}(\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack1, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 0, 1\bigr\rbrack, \bigl\lbrack0, 1, 1, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

Explicit vertex names replace the integers

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In$\mathrm{VertexList}(\mathrm{AdjacencyGraph}(\bigl\lbrack a, b, c, d\bigr\rbrack, \bigl\lbrack\bigl\lbrack1, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 0, 1\bigr\rbrack, \bigl\lbrack0, 1, 1, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack a, b, c, d\bigr\rbrack$

Five undirected edges on four vertices: $K_4$ minus an edge

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 0, 1\bigr\rbrack, \bigl\lbrack0, 1, 1, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(3, 4)\bigr\rbrack)$

All off-diagonal ones give the complete graph $K_5$

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 1, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 1, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 0, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 1, 1, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(1, 4), \mathrm{UndirectedEdge}(1, 5), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(2, 5), \mathrm{UndirectedEdge}(3, 4), \mathrm{UndirectedEdge}(3, 5), \mathrm{UndirectedEdge}(4, 5)\bigr\rbrack)$

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 0, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{DirectedEdge}(1, 2), \mathrm{DirectedEdge}(2, 1), \mathrm{DirectedEdge}(3, 1), \mathrm{DirectedEdge}(3, 2), \mathrm{DirectedEdge}(4, 1), \mathrm{DirectedEdge}(4, 2)\bigr\rbrack)$

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In$\bigl\lbrack\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack1, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 0, 1, 1\bigr\rbrack, \bigl\lbrack1, 1, 0, 1\bigr\rbrack, \bigl\lbrack0, 1, 1, 0\bigr\rbrack\bigr\rbrack), \mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 0, 0\bigr\rbrack, \bigl\lbrack1, 0, 0, 0\bigr\rbrack, \bigl\lbrack1, 1, 1, 0\bigr\rbrack, \bigl\lbrack1, 1, 0, 0\bigr\rbrack\bigr\rbrack)\bigr\rbrack$Out$\bigl\lbrack\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 1), \mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(2, 4), \mathrm{UndirectedEdge}(3, 4)\bigr\rbrack), \mathrm{Graph}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack\mathrm{DirectedEdge}(1, 2), \mathrm{DirectedEdge}(2, 1), \mathrm{DirectedEdge}(3, 1), \mathrm{DirectedEdge}(3, 2), \mathrm{DirectedEdge}(3, 3), \mathrm{DirectedEdge}(4, 1), \mathrm{DirectedEdge}(4, 2)\bigr\rbrack)\bigr\rbrack$

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack0, 1, 1, 1, 1\bigr\rbrack, \bigl\lbrack0, 0, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 0, 0, 0\bigr\rbrack, \bigl\lbrack0, 0, 0, 0, 0\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack, \bigl\lbrack\mathrm{DirectedEdge}(1, 2), \mathrm{DirectedEdge}(1, 3), \mathrm{DirectedEdge}(1, 4), \mathrm{DirectedEdge}(1, 5)\bigr\rbrack)$

Properties #

Diagonal ones become self-loops

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In$\mathrm{AdjacencyGraph}(\bigl\lbrack\bigl\lbrack1, 1, 1\bigr\rbrack, \bigl\lbrack1, 0, 1\bigr\rbrack, \bigl\lbrack1, 1, 1\bigr\rbrack\bigr\rbrack)$Out$\mathrm{Graph}(\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack\mathrm{UndirectedEdge}(1, 1), \mathrm{UndirectedEdge}(1, 2), \mathrm{UndirectedEdge}(1, 3), \mathrm{UndirectedEdge}(2, 3), \mathrm{UndirectedEdge}(3, 3)\bigr\rbrack)$

About

Examples	12
Checked against	Wolfram Language 12 agree
Crosswalk	Wolfram Language

See also: AdjacencyMatrix, Graph